State estimation in quantum homodyne tomography with noisy data

In the framework of noisy quantum homodyne tomography with efficiency parameter 0 < η ⩽ 1, we propose two estimators of a quantum state whose density matrix elements ρ m , n decrease as , for fixed known B > 0 and 0 < r ⩽ 2. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for 0 < η ⩽ 1/2), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in risk.In the framework of noisy quantum homodyne tomography with efficiency parameter 0 < η ⩽ 1, we propose two estimators of a quantum state whose density matrix elements ρ m , n decrease as , for fixed known B > 0 and 0 < r ⩽ 2. The first procedure estimates the matrix coefficients by a projection method on the pattern functions (that we introduce here for 0 < η ⩽ 1/2), the second procedure is a kernel estimator of the associated Wigner function. We compute the convergence rates of these estimators, in risk.

arXiv:0804.2434Inverse ProblemsNov 13, 200813 citations

Quantum Tomography under Prior Information

Teiko Heinosaari, Luca Mazzarella, Michael M. Wolf

arXiv:1109.5478Communications in Mathematical PhysicsFeb 4, 2013141 citations

Adaptive estimation of the density matrix in quantum homodyne tomography with noisy data

Pierre Alquier, Katia Méziani, Gabriel Peyré A

In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < η\leq 1$ , we propose a novel estimator of a quantum state whose density matrix elements $ρ_{m,n}$ decrease like $Ce^{-B(m+n)^{r/ 2}}$ , for fixed $C\geq 1$ , $B>0$ and $0In the framework of noisy quantum homodyne tomography with efficiency parameter $1/2 < η\leq 1$, we propose a novel estimator of a quantum state whose density matrix elements $ρ_{m,n}$ decrease like $Ce^{-B(m+n)^{r/ 2}}$, for fixed $C\geq 1$, $B>0$ and $0

arXiv:1301.7644Inverse ProblemsJun 24, 20136 citations

Nonparametric estimation of the purity of a quantum state in quantum homodyne tomography with noisy data

Katia Méziani

The aim of this work is to estimate a quadratic functional of a unknown Wigner function from noisy tomographic data. The Wigner function can be seen as the representation of the quantum state of a light beam. The estimation of a quadratic functional is done from result of quantum homodyne measurement performed on identically prepared quantum systems. We start by constructing an estimator of a quadratic functional of the Wigner function. We show that the proposed estimator is optimal or nearly optimal in a minimax sense over a class of infinitely differentiable functions. Parametric rates are also reached for some values of the smoothness parameters and the asymptotic normality is given. Then, we construct an adaptive estimator that does not depend on the smoothness parameters and prove it is minimax over some set-ups.The aim of this work is to estimate a quadratic functional of a unknown Wigner function from noisy tomographic data. The Wigner function can be seen as the representation of the quantum state of a light beam. The estimation of a quadratic functional is done from result of quantum homodyne measurement performed on identically prepared quantum systems. We start by constructing an estimator of a quadratic functional of the Wigner function. We show that the proposed estimator is optimal or nearly optimal in a minimax sense over a class of infinitely differentiable functions. Parametric rates are also reached for some values of the smoothness parameters and the asymptotic normality is given. Then, we construct an adaptive estimator that does not depend on the smoothness parameters and prove it is minimax over some set-ups.

arXiv:math/0611512Nov 16, 20060 citations

Homodyne state tomography with photon number resolving detectors

Graciana Puentes, Jeff S. Lundeen, Branderhorst +3 more

Conference on Lasers and Electro-OpticsJan 1, 20080 citations

In the framework of noisy quantum homodyne tomography with efficiency parameter

1/2 < η\leq 1

, we propose a novel estimator of a quantum state whose density matrix elements

ρ_{m,n}

decrease like

Ce^{-B(m+n)^{r/ 2}}

, for fixed

C\geq 1

B>0

and $0